The Appendix presents the theorems used in Section REF . All the fittings are based on the MFPC decomposition objects \(\widehat{\mathbf {\theta }}\) , so we omit \(\mathbf {\theta }\)  here for simplification. Given Propositions 1-7 from [1]} ([1]}), we assume \(\rho _{i,m}\)  and \(\mathbf {\epsilon }_i\)  are jointly Gaussian. Consider \(\forall j \in \lbrace 1,\ldots ,p\rbrace \) , \(\mathbf {t}:=(t^{(1)},\ldots ,t^{(p)}) \in \mathcal {T}\) , \({\mathbf {X}}_i(\mathbf {t}):=({X}^{(1)}_i(t^{(1)}),\ldots ,{X}^{(p)}_i(t^{(p)}))^\top \in \mathbb {R}^{p}\) , and \(\widehat{\mathbf {X}}_{i}(\mathbf {t}):=(\widehat{X}^{(1)}_{i}(t^{(1)}),\ldots ,\widehat{X}^{(p)}_{i}(t^{(p)}))^\top \in \mathbb {R}^{p}\) . Define \(\mathbf {\rho }_{iM}=(\rho _{i,1},\ldots ,\rho _{i,M})^\top \) , and \(\widehat{\mathbf {\rho }}_{iM}=(\widehat{\rho }_{i,1},\ldots ,\widehat{\rho }_{i,M})^\top \) ,
\({\mathbf {X}}_i(\mathbf {t})=\mathbf {\mu }(\mathbf {t})+\sum _{m=1}^{\infty } \rho _{i,m} \mathbf {\psi }_{i,m}(\mathbf {t})\) , and \(\vert \vert \vert \mathbf {X} \vert \vert \vert :=\sum _{j=1}^p\Vert X^{(j)}\Vert \) .
